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Interval Estimation Topic 16 Interval Estimation Our strategy to estimation thus far has been to use a method to find an estimator, e.g., method of moments, or maximum likelihood, and evaluate the quality of the estimator by evaluating the bias and the variance of the estimator. Often, we know more about the distribution of the estimator and this allows us to take a more comprehensive statement about the estimation procedure. Interval estimation is an alternative to the variety of techniques we have examined. Given data x, we replace the point estimate ✓ˆ(x) for the parameter ✓ by a statistic that is subset Cˆ(x) of the parameter space. We will consider both the classical and Bayesian approaches to choosing Cˆ(x) . As we shall learn, the two approaches have very different interpretations. 16.1 Classical Statistics In this case, the random set Cˆ(X) is chosen to have a prescribed high probability, γ, of containing the true parameter value ✓. In symbols, P ✓ Cˆ(X) = γ. ✓{ 2 } In this case, the set Cˆ(x) is called a γ-level con- fidence set. In the case of a one dimensional param- 0.4 eter set, the typical choice of confidence set is a con- fidence interval 0.35 Cˆ(x)=(✓ˆ`(x), ✓ˆu(x)). 0.3 Often this interval takes the form 0.25 Cˆ(x)=(✓ˆ(x) m(x), ✓ˆ(x)+m(x)) = ✓ˆ(x) m(x) − ± where the two statistics, 0.2 ✓ˆ(x) is a point estimate, and • 0.15 m(x) is the margin of error. • 0.1 16.1.1 Means 0.05 area α Example 16.1 (1-sample z interval). If X1.X2....Xn 0 µ z are normal random variables with unknown mean −3 −2 −1 0 1 2 α 3 2 but known variance σ0. Then, . ↵ is the area under the standard X¯ µ Figure 16.1: Upper tail critical values Z = − normal density and to the right of the vertical line at critical value z↵ σ0/pn 239 Introduction to the Science of Statistics Interval Estimation is a standard normal random variable. For any ↵ between 0 and 1, let z↵ satisfy P Z>z = ↵ or equivalently P Z z =1 ↵. { ↵} { ↵} − The value is known as the upper tail probability with critical value z↵. We can compute this in R using, for example > qnorm(0.975) [1] 1.959964 for ↵ =0.025. If γ =1 2↵, then ↵ =(1 γ)/2. In this case, we have that − − P z <Z<z = γ. {− ↵ ↵} Let µ0 is the state of nature. Taking in turn each the two inequalities in the line above and isolating µ0, we find that X¯ µ0 − = Z<z↵ σ0/pn X¯ µ <z σ0 − 0 ↵ pn σ X¯ z 0 <µ − ↵ pn 0 Similarly, X¯ µ0 − = Z> z↵ σ0/pn − implies σ µ < X¯ + z 0 0 ↵ pn Thus σ σ X¯ z 0 <µ < X¯ + z 0 . − ↵ pn 0 ↵ pn has probability γ. Thus, for data x, σ0 x¯ z(1 γ)/2 ± − pn is a confidence interval with confidence level γ. In this case, µˆ(x)=¯x is the estimate for the mean and m(x)=z(1 γ)/2σ0/pn is the margin of error. − We can use the z-interval above for the confidence interval for µ for data that is not necessarily normally dis- tributed as long as the central limit theorem applies. For one population tests for means, n>30 and data not strongly skewed is a good rule of thumb. Generally, the standard deviation is not known and must be estimated. So, let X ,X , ,X be normal random 1 2 ··· n variables with unknown mean and unknown standard deviation. Let S2 be the unbiased sample variance. If we are forced to replace the unknown variance σ2 with its unbiased estimate s2, then the statistic is known as t: x¯ µ t = − . s/pn The term s/pn which estimates the standard deviation of the sample mean is called the standard error. The remarkable discovery by William Gossett is that the distribution of the t statistic can be determined exactly. Write pn(X¯ µ) Tn 1 = − . − S Then, Gossett was able to establish the following three facts: 240 Introduction to the Science of Statistics Interval Estimation 0.4 1.0 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 -4 -2 0 2 4 -4 -2 0 2 4 x x Figure 16.2: The density and distribution function for a standard normal random variable (black) and a t random variable with 4 degrees of freedom (red). The variance of the t distribution is df / (df 2) = 4/(4 2) = 2 is higher than the variance of a standard normal. This can be seen in the − − broader shoulders of the t density function or in the smaller increases in the t distribution function away from the mean of 0. The numerator is a standard normal random variable. • The denominator is the square root of • 1 n S2 = (X X¯)2. n 1 i − i=1 − X The sum has chi-square distribution with n 1 degrees of freedom. − The numerator and denominator are independent. • With this, Gossett was able to compute the density of the t distribution with n 1 degrees of freedom. Gossett, − who worked for the the brewery of Arthur Guinness in Dublin, was permitted to publish his results only if it appeared under a pseudonym. Gosset chose the name Student, thus the distribution is sometimes known as Student’s t. Again, for any ↵ between 0 and 1, let upper tail probability tn 1,↵ satisfy − P Tn 1 >tn 1,↵ = ↵ or equivalently P Tn 1 tn 1,↵ =1 ↵. { − − } { − − } − We can compute this in R using, for example > qt(0.975,12) [1] 2.178813 for ↵ =0.025 and n 1 = 12. − Example 16.2. For the data on the lengths of 200 Bacillus subtilis, we had a mean x¯ =2.49 and standard deviation s =0.674. For a 96% confidence interval ↵ =0.02 and we type in R, > qt(0.98,199) [1] 2.067298 241 Introduction to the Science of Statistics Interval Estimation 4 3 2 1 0 10 20 30 40 50 df Figure 16.3: Upper critical values for the t confidence interval with γ =0.90 (black), 0.95 (red), 0.98 (magenta) and 0.99 (blue) as a function of df , the number of degrees of freedom. Note that these critical values decrease to the critical value for the z confidence interval and increases with γ. Thus, the interval is 0.674 2.490 2.0674 =2.490 0.099 or (2.391, 2.589) ± p200 ± Example 16.3. We can obtain the data for the Michaelson-Morley experiment using R by typing > data(morley) The data have 100 rows - 5 experiments (column 1) of 20 runs (column 2). The Speed is in column 3. The values for speed are the amounts over 299,000 km/sec. Thus, a t-confidence interval will have 99 degrees of freedom. We can see a histogram by writing hist(morley$Speed). To determine a 95% confidence interval, we find > mean(morley$Speed) [1] 852.4 > sd(morley$Speed) [1] 79.01055 > qt(0.975,99) [1] 1.984217 Thus, our confidence interval for the speed of light is 79.0 299, 852.4 1.9842 = 299, 852.4 15.7 or the interval (299836.7, 299868.1) ± p100 ± This confidence interval does not include the presently determined values of 299,792.458 km/sec for the speed of light. The confidence interval can also be found by tying t.test(morley$Speed). We will study this command in more detail when we describe the t-test. 242 Introduction to the Science of Statistics Interval Estimation Histogram of morley$Speed 30 25 20 15 Frequency 10 5 0 600 700 800 900 1000 1100 morley$Speed Figure 16.4: Measurements of the speed of light. Actual values are 299,000 kilometers per second plus the value shown. Exercise 16.4. Give a 90% and a 98% confidence interval for the example above. We often wish to determine a sample size that will guarantee a desired margin of error. For a γ-level t-interval, this is s m = tn 1,(1 γ)/2 . − − pn Solving this for n yields 2 tn 1,(1 γ)/2 s n = − − . m ✓ ◆ Because the number of degrees of freedom, n 1, for the t distribution is unknown, the quantity n appears on both − sides of the equation and the value of s is unknown. We search for a conservative value for n, i.e., a margin of error that will be no greater that the desired length. This can be achieved by overestimating tn 1,(1 γ)/2 and s. For the speed of light example above, if we desire a margin of error of m = 10 km/sec for a 95% confidence− − interval, then we set tn 1,(1 γ)/2 =2and s = 80 to obtain − − 2 80 2 n · = 256 ⇡ 10 ✓ ◆ measurements are necessary to obtain the desired margin of error.. The next set of confidence intervals are determined, in the case in which the distributional variance in known, by finding the standardized score and using the normal approximation as given via the central limit theorem. In the cases in which the variance is unknown, we replace the distribution variance with a variance that is estimated from the observations. In this case, the procedure that is analogous to the standardized score is called the studentized score.
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